Is rather a question of calculus skills, but how do I get the time derivative of the Hubble parameter here in [1]? Is it the Leibnitz rule, the chain rule, some clever re-arrangement?

thank you

equation [1] is a straightforward application of the definition of H(t) = a'/a
and the Leibnitz rule
or I would call it the "quotient rule" for taking derivative of f(t)/g(t)

You can also think of it as the "product rule" applied to the two functions f(t) and (1/g(t))

Notice that (1/a)' = (-a'/a2) (I guess you could call that an application of "chain rule")

so you just make a simple application of product rule to H(t) = a' * (1/a)

and you get a'' * (1/a) + a' * (-a'/a2) = a''/a - (a'/a)2

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Be careful there is something misleading in the last 4 or 5 equations on that page you linked to!

In the standard cosmological model the derivative of H is always negative.

That page is a bit inconsistent because it introduces the Friedman equations WITH LAMBDA THE COSMOLOGICAL CONSTANT, so there should be no "dark energy component" in the energy density rho, and the pressure p.

rho and p are just positive terms like you would expect, nothing tricky.

So he actually shows that the derivative of H is NEGATIVE because it is proportional to
-(rho + p), which is negative.

And this is correct, according to standard model (where you have a cosmological constant).

As matter thins out, rho + p goes to zero,

so the derivative of H, which is negative, goes to zero. So H declines slower and slower and levels out to a constant positive percentage growth rate H∞ in the limit.

This is what is meant by "accelerated expansion" (H declining to a positive limiting value) because growth at a constant percentage rate is, of course, EXPONENTIAL growth. So if you watch a particular distance grow, it goes like money in the bank at a constant percentage rate of interest.

"Acceleration" does not mean that H(t) should increase. In the standard model with cosmo constant Lambda, it just means that the DECLINE of H(t) is leveling out to a small positive value so we get exponential growth.

Currently H is about 1/144 of a percent per million years, and the expected H∞ limit is 1/173 of a percent per million years.